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User blog:Ikosarakt1/Variant of BEAF-definitions for up-arrow notation-notation level.
I found a new, quite easy way of defining structures for BEAF up to \(X \uparrow^X X\). The definition goes as follows: Rule 1. (n=1) A{n}B = A^B Rule 2. (B=0) A{n}B = 1 Rule 3. (B doesn't contain X's) A{n+1}(B+1) = A{n}(A{n+1}B) Rule 4. (otherwise) A{n+1}(B+1) = (A{n+1}B){n}X Note that it works as strong hyper-operators when B is just a number, and weak hyper-operators when B contains X's. It means that, for example, \(X\{2\}5 = X^{X\{2\}4}\) while \(X\{2\}(X+1) = (X\{2\}X)^X\). Note that "true" solution of \(X\{2\}(X+1)\), supposed by Bowers, is \(X^{X\{2\}X}\), but then we don't know how to evaluate it further. Structural comparisons with traditional ordinal notations While "Rule 4" gives us a system which certainly weaker than the hypothetical which works as pure right-associative, they grow at the same rate when we diagonalize over them. It think the following comparisons are correct: X{2}X = \(\varepsilon_0\) (X{2}X){1}2 = \(\varepsilon_0^2 = \omega^{\varepsilon_0*2}\) (X{2}X){1}3 = \(\varepsilon_0^3 = \omega^{\varepsilon_0*3}\) X{2}(X+1) = \(\varepsilon_0^\omega = \omega^{\omega^{\varepsilon_0+1}}\) X{2}(X+2) = \(\varepsilon_0^{\omega^2} = \omega^{\omega^{\varepsilon_0+2}}\) X{2}(X*2) = \(\varepsilon_0^{\omega^\omega} = \omega^{\omega^{\varepsilon_0+\omega}}\) X{2}(X{2}X) = \(\varepsilon_0^{\varepsilon_0} = \omega^{\omega^{\varepsilon_0*2}}\) X{2}(X{2}(X+1)) = \(\varepsilon_0^{\varepsilon_0^\omega} = \omega^{\omega^{\omega^{\varepsilon_0+1}}}\) X{2}(X{2}(X{2}X)) = \(\varepsilon_0^{\varepsilon_0^{\varepsilon_0}} = \omega^{\omega^{\omega^{\varepsilon_0*2}}}\) X{3}X = \(\varepsilon_1\) So it turned out that X^^^X by that definition is just at level of \(\varepsilon_1\). Don't be fooled however, believing that X^^^^X will be at level of just \(\varepsilon_2\). Just look at further behavior: (X{3}X){1}X = \(\varepsilon_1^\omega = \omega^{\omega^{\varepsilon_1+1}}\) (X{3}X){1}(X{3}X) = \(\varepsilon_1^{\varepsilon_1} = \omega^{\omega^{\varepsilon_1*2}}\) (X{3}X){1}(X{3}X){1}(X{3}X) = \(\varepsilon_1^{\varepsilon_1^{\varepsilon_1}} = \omega^{\omega^{\omega^{\varepsilon_1*2}}}\) (X{3}X){2}X = X{3}(X+1) = \(\varepsilon_2\) (X{3}(X+1)){2}X = X{3}(X+2) = \(\varepsilon_3\) X{3}(X+n) = \(\varepsilon_{n+1}\) X{3}(X*2) = \(\varepsilon_\omega\) X{3}(X{2}X) = \(\varepsilon_{\varepsilon_0}\) X{3}(X{3}X) = \(\varepsilon_{\varepsilon_1}\) X{4}X = \(\zeta_0\) Consider that if A has level \(\alpha\), then A{2}X has level \(\varepsilon_{\alpha+1}\): (X{4}X){2}X = \(\varepsilon_{\zeta_0+1}\) (X{4}X){2}(X+1) = \(\omega^{\omega^{\varepsilon_{\zeta_0+1}+1}}\) (X{4}X){2}(X{4}X) = \(\omega^{\omega^{\varepsilon_{\zeta_0+1}*2}}\) (X{4}X){3}X = X{4}(X+1) = \(\varepsilon_{\zeta_0+2}\) X{4}(X+2) = \(\varepsilon_{\zeta_0+3}\) X{4}(X*2) = \(\varepsilon_{\zeta_0+\omega}\) X{4}(X{4}X) = \(\varepsilon_{\zeta_0*2}\) X{4}(X{4}(X{4}X)) = \(\varepsilon_{\varepsilon_{\zeta_0*2}}\) X{5}X = \(\zeta_1\) We see that each adding to B in A{3}B (when B contains X's) leads to the level of new epsilon number. (X{5}X){2}X = \(\varepsilon_{\zeta_1+1}\) (X{5}X){3}X = \(\varepsilon_{\zeta_1+2}\) (X{5}X){3}(X+1) = \(\varepsilon_{\zeta_1+3}\) (X{5}X){3}(X+n) = \(\varepsilon_{\zeta_1+n+2}\) (X{5}X){3}(X{5}X) = \(\varepsilon_{\zeta_1*2}\) (X{5}X){3}(X{5}X){3}(X{5}X) = \(\varepsilon_{\varepsilon_{\zeta_1*2}}\) (X{5}X){4}X = X{5}(X+1) = \(\zeta_2\) So "hexating" A in this system will lead to the level \(\zeta_{\alpha+1}\) X{5}(X+1) = \(\zeta_3\) X{5}(X+2) = \(\zeta_4\) X{5}(X{5}X) = \(\zeta_{\zeta_0}\) X{6}X = \(\eta_0\) Consider the sequence: X{2}X = \(\varepsilon_0 = \varphi(1,0)\) X{3}X = \(\varepsilon_1 = \varphi(1,1)\) X{4}X = \(\zeta_0 = \varphi(2,0)\) X{5}X = \(\zeta_0 = \varphi(2,1)\) X{6}X = \(\eta_0\) Using the pattern, we conclude that X{2n}X = \(\varphi(n,0)\) and X{2n+1}X = \(\varphi(n,1)\). So we indeed reach the power of \(\varphi(\omega,0)\). Pros and cons Pros: *Easy to define and doesn't need additional notation like explained here. *Exhibits the same growth rate as hypothetical "normal" variant of BEAF. Cons: *General avoiding the principle of BEAF: when in A & n, A composed with X's and operations, the total number of entries can be found just replacing X's to n's and solving. When we solve X{2}(X+1), we expect\(3^{3^{3^3}}\) entries, but replacing the first expression to (X{2}X){1}X we get instead only \(3^{3^3+1}\) of them. *Definition of increment to B in up-arrow notation is dependent on existence of X's. So the pattern breaks on base-recursing and polyponent-recursing. It looks awkwardly. Category:Blog posts